Scaling of finite size effect of $\alpha$-R\'enyi entropy in disjointed intervals under dilation

TL;DR

This paper investigates the finite size effects of $ ext{alpha}$-Rényi entropy in disjoint intervals of the XY model, revealing a universal scaling law and distinguishing between intrinsic and extrinsic effects, with implications for understanding entanglement in many-body systems.

Contribution

It introduces a universal scaling law for the finite size effect of $ ext{alpha}$-Rényi entropy in disjoint intervals under dilation, extending understanding beyond single-interval cases.

Findings

Discovered a universal scaling law for FSE in disjoint intervals.

Identified two types of FSE: intrinsic and extrinsic.

Edge modes at open ends contribute predominantly to FSE.

Abstract

The $\alpha$-R\'enyi entropy in the gapless models have been obtained by the conformal field theory, which is exact in the thermodynamic limit. However, the calculation of its finite size effect (FSE) is challenging. So far only the FSE in a single interval in the XX model has been understood and the FSE in the other models and in the other conditions are totally unknown. Here we report the FSE of this entropy in disjointed intervals $A = \cup_i A_i$ under a uniform dilation $\lambda A$ in the XY model, showing of a universal scaling law as \begin{equation*} \Delta_{\lambda A}^\alpha = \Delta_A^\alpha \lambda^{-\eta} \mathcal{B}(A, \lambda), \end{equation*} where $|\mathcal{B}(A, \lambda)| \le 1$ is a bounded function and $\eta = \text{min}(2, 2/\alpha)$ when $\alpha < 10$. We verify this relation in the phase boundaries of the XY model, in which the different central charges correspond to the physics of free Fermion and free Boson models. We find that in the disjointed intervals, two FSEs, termed as extrinsic FSE and intrinsic FSE, are required to fully account for the FSE of the entropy. Physically, we find that only the edge modes of the correlation matrix localized at the open ends $\partial A$ have contribution to the total entropy and its FSE. Our results provide some incisive insight into the entanglement entropy in the many-body systems.